# Number System-CAT

**The remainder obtained when a prime number greater than 6 is divided by 6 is [1995]**

**a. 1 or 3**

**b. 1 or 5**

**c. 3 or 5**

**d. 4 or 5**

**5^6-1 is divisible by**

** a. 5 b.13 c. 31 d. None of these. (1995)**

**Two positive integers differ by 4 and sum of their reciprocals is 10/21, then one of the numbers is a. 3 **

**b. 1 **

**c. 5 **

**d. 21 [1995]**

**Three consecutive positive even numbers are such that thrice the first number exceeds double the third by 2, then the third number is [1995].**

**a. 10**

**b. 14**

**c. 16**

**d. 12**

**If a number 774958A96B is to be divisible by 8 and 9, the respective values of A and B will be [1996].**

** a. 7 and 8**

** b. 8 and 0**

** c. 5 and 8**

** d. 3 and 6**

**If m and n are integers divisible by 5, which of the following is not necessarily true? [1997]**

**Consider a sequence, where the nth term is tn = n/n+2. The value of t3 * t4 * t5 * … t53 equals(2000)**

**A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour? **

**a. 30 b. 24 c. 20 d. 60**

**[CAT, 2001]**

**7**^{6n}**- 6**^{6n}**, where n is an integer >0, is divisible by**

**a 13 b. 127 c.. 559 d. All of these **

**[CAT, 2002]**

**A child was asked to add first few natural numbers (that is, 1 + 2 + 3......) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong the child discovered he had missed one number in the sequence during addition. The number he missed was:**

**a. less than 10 **

**b. 10 **

**c. 15 **

**d. more than 15 [CAT, 2002]**

**What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7? **

**[CAT, 2003]**

**(a) 666 **

**(b) 683 **

**(c) 676 **

**(d) 777 **

**What is the remainder when 4**^{96 }**is divided by 6 ? [CAT, 2003]**

**a. 0 **

**b. 2 **

**c. 3 **** **

**d. 4 [CAT, 2003]**

**Which among 2^1/2, 3^1/3, 4^1/4, 12^/12 is the largest? (2004)**

** ****The remainder, when 15^23 23^23 is divided by 19 is?**

**a.4 b.15 c.0 d.18 **

**[CAT, 2004]**

**Let S be a set of positive integers such that every element n of S satisfies the conditions**

**a) 1000 ≤ n ≤ 1200**

**b) Every digit in n is odd.**

**Then how many elements of S are divisible by 3? [2005]**

**a.9 **** **

**b.10 **

**c.11 **

**d.12 **

**When you reverse the digits of the number 13, the number increases by 18. How many other 2 digit numbers increase by 18 when their digits are reversed? [CAT, 2006]**

**a. 5 b. 6 c. 7 d. 8 e. 10**

**If [latex]\frac { a }{ b } =\frac { 1 }{ 3 } ,\frac { b }{ c } =2,\quad \frac { c }{ d } =\frac { 1 }{ 2 } ,\frac { d }{ e } =3[/latex] and [latex] \frac { e }{ f } =\frac { 1 }{ 4 }[/latex] then what is the value of [latex] \frac { abc }{ def }[/latex] ?**

**a. [latex]\frac { 3 }{ 8 }[/latex] **

**b. [latex]\frac { 27 }{ 8 }[/latex] **

**c. [latex]\frac { 3 }{4 }[/latex] **

** d. [latex]\frac { 27 }{ 4 }[/latex] **

**e. [latex]\frac { 1 }{ 4 }[/latex] **

**Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many are such numbers of perfect squares?**

**a. 2 **

**b. 4 **

**c. 0 **** **

**d. 1 [CAT, 2007]**

**How many pairs of positive Integers m, n satisfy , [latex] \frac { 1 }{ m } +\frac { 4 }{ n } =\frac { 1 }{ 12 }[/latex] where n is an odd integer less than 60? **

**a. 4 **

**b. 7 **

** c. 5 **

** d. 3 [CAT, 2007]**

**What is the remainder when 4**^{96 }**is divided by 6 ? [CAT, 2003]**

**a. 0 **

**b. 2 **

**c. 3 **

**d. 4 [CAT, 2003]**

**What is the remainder when 7**^{74 }**- 5**^{74}** is divided by 4.**

**a.0 **

**b. 1 **

**c. 2 **

**d. None of these [CAT, 2009]**

**In a tournament, there are n teams T1, T2 ….., Tn with n > 5. Each team consists of k players, k > 3. The following pairs of teams have one player in common: T1 & T2, T2 & T3,……, Tn− 1 & Tn, and Tn & T1. No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together? [CAT, 2007]**

**a. k(n-1) b. n(k-2) c. k(n-2) d. (n-1) (k-1) e. n(k-1)**

**a^2-ab b^2/a^2 ab b^2=1/3 then find a/b.**

**a. 1 b.2 c. 3 d. 4 [CAT, 2009]**

**Let S**_{n}** denote the sum of squares of the first n odd natural numbers. If S**_{n }**= 533n, find the value of n.**

**a. 18 **

**b. 20**

**c.24 **

** d. 30 [CAT, 2011]**

**The value of numbers 2**^{2004}** and 5**^{2004}** are written one after another. How many digits are there in all?**

**a. 2008 **

**b. 2003 **

**c.2004 **

** d. 2005 [CAT, 2011]**

**If p be a prime number, p>3 and let x be the product of positive number 1, 2, 3,…,(p-1), then consider the following statements**

**1. x is a composite number divisible by p.**

**2. x is a composite number not divisible by but some prime number greater than p may divide x.**

**3.x is not divisible by any prime number (p-2).**

**4. all prime numbers less than (p-1) divide x.**

**Which of the following statement is/are correct** **[CAT, 2012]**

**(a) I and II are correct **

**(b) II and III are correct**

**(c) III and IV are correct **

**(d) IV alone is correct**

**What is the sum of all two digit numbers which leave a remainder of 6 when divided by 8? [CAT, 2012]**

**(a) 612**

**(b) 594 **

**(c) 23 **

**(d) 872**

**Find the remainder of 2**^{1040}** divided by 131. [CAT, 2012]**

**(a) 1 **

**(b) 3 **

**(c) 5 **

**(d) 7**

**A three-digit number which on being subtracted from another three-digit number consisting of the same digits in reverse order gives 594. The minimum possible sum of all the three digits of this number is**

** [CAT, 2013]**

**(a) 6 **

**(b) 7**

**(c) 8**

**(d) cannot be determined**

If a1=1 and a(n 1)-3an 2=4n for every positive integer n, then a100 equals ? (2014)

**a. 3**^{99}**-200 **

**b. 3**^{99}**+200 **

**c. 3**^{100}**-200 **

**d. 3**^{100}**+200 [CAT, 2014]**

**Suppose n is an integer, such that the sum of the digits of n is 2, and 10**^{10}** < n < 10**^{11}**. The number of different values for n is:**

**a. 11 **

**b. 10 **

**c. 9 **

**d. 8 [CAT, 2014]**

A sequence of 4 digits, when considered as a number in base 10 is four times the number it represents in base 6. What is the sum of the digits of the sequence?

a. 7

b. 6

c. 9

d. 8 [CAT, 2016]

**What is the greatest power of 5 that divides 80! exactly.**

**a. 15 **

**b. 16 **

**c. 19 **

**d. 13 [CAT, 2016]**

**LHS = RHS (left hand side = right hand side)**

**Firstly notice that we have the bases same which is 9 in each three terms (LHS and RHS):**

**Let’s take the second term in LHS first = 9**^{2}**=81 and using this rule ( m**^{n}**)**^{p}** = m**^{np}**, we multiply p= (x-1) by n=2 i.e 2(x-1) = 2x-2.**

**Each family in a locality has at most two adults, and no family has fewer than three children. Considering all the families together there are more adults than boys, more boys than girls and more girls than families. Then the minimum possible number of families in the locality is: [2004].**

** a. 4 b.5 c. 2 d.3**

**Let x and y be positive integers such that x is prime and y is composite. Then**

**a. y - x can't be an even integer**

**b. yx can't be an even integer**

**c. (x+y)/x can't be even integer**

**d. None of these [2003]**

**If the sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms? [CAT, 2004]**

**a. 0 b. -1 c.1 d. Not Unique **

**The number of common terms in the two sequences 17, 21, 25, ..., 417 and 16, 21, 26, ..., 466 is **

**a. 19 b.20 c.77 d.78 (2008)**

**What is the remainder when 7 ^ 74 - 5 ^74 is divided by 4? (2008)**

### if a^2-ab b^2/a^2 ab b^2=1/3, then find a/b

**a.1 b.2 c.3 d. 4 (2009)**

**The largest number among the following that will perfectly divide 101^100-1 is.**

**a.100 b. 10000 c. 100^100** ** d. 100000 ** ** (2010) **

### How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4 if repetition of digits is allowed? a. 499 b. 500 c. 375 d. 376 (2010)

**Find the smallest number which when increased by 5 is exactly divisible by 8, 11 and 24. [CAT, 1994]**

**A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number. If his answer differed from the actual one by 770, find the number. [CAT, 1997] **

** a. 1584 b. 2520 c. 1728 d. 1656**

**If n**^{3}** is odd, which of the following statements are true? [CAT, 1998] **

** I. n is odd. **

**II. n**^{2}** is odd. **

**III. n**^{2}** is even**

**What is the digit in the unit’s place of 2**^{51}**? [CAT, 1998]**

**a. 2 b. 8 c. 1 d. 4**

*If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? [CAT, 1999]*

*If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? [CAT, 1999]*

* A. N is odd B. n is prime C. n is a perfect square *

*A. N is odd B. n is prime C. n is a perfect square*

* a. A and C only b. A and B only c. A only d. None of these*

*a. A and C only b. A and B only c. A only d. None of these*

**The integers 34041 and 32506, when divided by a three-digit integer n, leave the same remainder. What is the value of n? [CAT, 2000]**

**(a) 289 (b) 367 (c) 453 (d) 307**

*Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product? [CAT, 2001]*

*Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product? [CAT, 2001]*

*a. 1050 b. 540 c.1440 d.1590*

*a. 1050 b. 540 c.1440 d.1590*